Abstract
In this paper I describe how several notions and constructions in topos logic can be dualized, giving rise to complement-toposes with their paraconsistent internal logic, instead of the usual standard toposes with their intuitionistic logic.
For Christian Edward Mortensen, long-distance mentor, in his 70th birthday.
This paper is conceptually the first one of a tetralogy gathering up my logico-philosophical investigations on topos logic, deeply motivated by the philosophy behind Universal Logic: The other papers are [12, 15, 16], in that order. However, insofar as each paper is self-contained for it can be read independently, the reader might find several similarities between them in their introductions, speaking of the general motivation, or their presentations of the basics of topos logic, but each deals with special, specific problems. This paper has been written under the support from the CONACyT project CCB 2011 166502 “Aspectos filosóficos de la modalidad,” as well as from the PAPIIT project IA401015 “Tras las consecuencias. Una visión universalista de la lógica (I).” I thank Axel Barceló-Aspeitia, Jean-Yves Béziau, Carlos César Jiménez, Chris Mortensen, Zbigniew Oziewicz, Ivonne Pallares-Vega and now countless referees for their comments, suggestions or encouragement over the years I have spent working on this topic. Diagrams were drawn using Paul Taylor’s diagrams package v. 3.94.
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Notes
- 1.
I use the word ‘slogan’ here pretty much in the sense of van Inwagen: “a vague phrase of ordinary English whose use is by no means dictated by the mathematically formulated speculations it is supposed to summarize” [51, p. 163], “but that looks as if it was,” I would add.
- 2.
- 3.
A discussion of other topics in philosophy of logic, like the issue of meaning variance or the discussion of the connections between degenerate categories and trivialism, is certainly worth, but that is material for separate work.
- 4.
This elucidation of toposes in logical terms follows closely [1].
- 5.
As Awodey has noted, this is Russell’s notion of propositional function, for example in The Principles of Mathematics Sect. 22 or Principia Mathematica, pp. 14 and 161.
- 6.
Note by the way that, unlike many authors, I prefer the equalizers presentation of logic, not the pullbacks one.
- 7.
This is not as odd as it might seem at first sight; see [14].
- 8.
- 9.
I will omit operations on \(_{S}\Omega \) for simplicity, since it does not constitute a fundamental missing in the theme of the internal logic.
- 10.
For brevity, I often will talk only of “connectives,” since their standard character can be obviated in this chapter, and their arity and order will be made explicit only when needed.
- 11.
See for example [18, Sect. 6.6].
- 12.
I have made a little abuse of notation, for I used ‘\(_{S}p\)’ in both \(\models _{_{S}\mathcal {E}}\) and \(\models _{I}\). In rigor, \(_{S}p\) is a morphism which corresponds to a formula \((_{S}p)^{*}\) in a possibly different language, but there is no harm if one identifies them. A proof can be found in [18, see Sect. 8.3 for the soundness part and Sect. 10.6 for the completeness part].
- 13.
It must be said that some categories might not have enough structure to support some of the logical notions mentioned here. For example, an object in a category might have no minimal subobject distinct from the maximal one (so no proposition always false is distinct from always true) or might have no coproducts (and therefore, would lack disjunctions), etc.
- 14.
The difference is clearly explained in [18, Sect. 7.3].
- 15.
For more on this, see Sect. 18.3.
- 16.
In Hegel’s Science of Logic, logic is divided into two parts, one of which is the logic of Being, the objective logic. Being is thought of here as an outer world beyond any particular subjective mind but still conceptually informed (and thus with a logic): It is “objectified spirit.” The other part is the logic of thinking, the subjective logic (Hegel calls it thus because thinking requires a thinking subject, more or less like a subjective right is the right of a subject). Hegel’s subjective logic is what today (and also in Hegel’s times) is commonly called ‘logic.’ An excellent treatment of the connections between Lawvere’s and Hegel’s views on logic can be found in [47]. Further clarification of the term ‘objective’ as applied to logic by Lawvere comes also from Hegel, but not from his work in logic but from his work on ethics, especially in Philosophy of Right (intended to be read with his Science of Logic as background), which is essentially a developed version of the section “Objective Spirit” in the Encyclopaedia of the Philosophical Sciences’ Philosophy of Spirit. According to Hegel, the objective spirit consists of collective, social practices, whereas the subjective spirit is the individual. Part of Lawvere’s “historical and dialectical realism” is that mathematical entities exist objectively, like being in the Logic, but that existence is determined in mathematical experience as a whole, including its collective practice (like the objective spirit)—not in a Platonic realm independent of any subject, nor in merely subjective, individual, experience.
- 17.
For simplicity I will omit quantifiers here and in the following section.
- 18.
For more details on what follows, see [31].
- 19.
These constructions are mathematical manifestations of the double-negation translation of classical zero-order logic into intuitionistic zero-order logic which works by inserting \(\lnot \lnot \) in relevant places. See the first section of [9] for more about these translations.
- 20.
It is important to set their individual contributions. Of the ten diagrams in [39, Chap. 11], Mortensen drew the first one and the final five, while Lavers drew the remaining four. The diagram for the dual-conditional never was explicitly drawn, but it was discussed in [39, p. 109]. The full story, as told by Mortensen in personal communication, is as follows. Mortensen gave a talk at the Australian National University (Canberra) in late 1986, on paraconsistent topos logic, arguing the topological motivation for closed set logic. He defined a complement-topos, drew the first three diagrams from Inconsistent Mathematics, Chap. 11, that is including the complement versions of \(_{S}true\) and paraconsistent negation, and criticized Goodman’s views on the conditional. But it was not seen clearly at that stage how the logic would turn out. Peter Lavers was present (also Richard Routley, Robert K. Meyer, Michael A. McRobbie, Chris Brink and others). For a couple of days in Canberra, Mortensen and Lavers tried without success to thrash it out. Mortensen returned home to Adelaide and two weeks later Lavers’ letter arrived in Adelaide, in which he stressed that inverting the order is the key insight to understanding the problem, drew the diagrams for conjunction and disjunction, and pointed out that subtraction is the right topological dual for the conditional. Mortensen then responded with the four diagrams for the S5 conditional, and one for quantification (last five diagrams in Inconsistent Mathematics, Chap. 11). A few months later (1987) Mortensen wrote the first paper, with Lavers as co-author, and sent it to Saunders Mac Lane and William Lawvere (also Routley, Meyer, Priest). Mac Lane replied but Lawvere did not. A later version of that paper became the eleventh chapter of Inconsistent Mathematics. I thank Prof. Mortensen for providing me this information.
- 21.
Mortensen and Lavers use the names complement-classifier and complement-topos, which are now the names set in the literature (cf. [13, 39, 40, 52]). Although the use of ‘dual topos’ would be appropriate here but misleading given the usual understanding of ‘dual category,’ I think there is no similar problem for the classifier of a complement-topos.
- 22.
The same remark on note 7 in Sect. 18.2 applies here.
- 23.
Again, for brevity I often will talk only of “connectives,” since their dual character can be obviated, and their arity and order will be made explicit only when needed.
- 24.
This is a controversial point. Mortensen thinks that functionality is mathematically prior to, and a more important matter than some logical notions. Someone might object to this by saying that ordinary math books use conditional constantly, and for example use definitions stated as conditionals so that one constantly quantifies into conditional contexts, and every time one proves that some object does not have a defined property one is negating a conditional. Seemingly, this cannot all be pushed into the metalanguage without severe contortions.
- 25.
I have attempted such a categorial description of this kind of duality in [12].
- 26.
Again, I have made a little abuse of notation, for I used ‘\(_{D}p\)’ in both \(\models _{_{D}\mathcal {E}}\) and \(\models _{I}\). In rigor, \(_{D}p\) is a morphism which corresponds to a formula \((_{D}p)^{*}\) in a possibly different language.
- 27.
By abuse of notation but to simplify reading I will not indicate that the order here is dual to that in standard toposes, unless there is risk of confusion.
- 28.
- 29.
Thus, as Vasyukov [52, p. 292] points out: “(...) in Set we always have paraconsistency because of the presence of both types of subobject classifiers (...)” just as we always have in it (at least) intuitionistic logic. The presence of paraconsistency within classical logic is not news. See for example [7], where some paraconsistent negations in S5 and classical first-order logic are defined.
- 30.
It is easy to verify that after making all the necessary changes, i.e., changing \(_{S}true_{{S}^{\downarrow \downarrow }}\) for \(_{D}{} \textit{false}_{{S}^{\downarrow \downarrow }}\), etc., the names are ordered in the same way as they are in \(_{S}{{S}^{\downarrow \downarrow }}\).
- 31.
A dualization of Kripke semantics for intuitionistic logic similar to that presented here was studied in [48]. Shramko maintains the Popperian reading to provide a “logic of refutation”; the crucial distinction lies in the condition for \(\sim \). Moreover, Shramko, like Goodman, also omits the discussion of subtraction. Goré [20, p. 252] even says that “[Goodman] annoyingly fails to give the crucial clause for satisfiability for his ‘pseudo-difference’ connective.”
- 32.
An interior operator is an operator which is multiplicative, idempotent, and deflationary, i.e., \(x\ge jx\). A closure operator is additive (\(j(x\cup y) = jx\cup jy\)), idempotent and inflationary.
- 33.
- 34.
As I have said before, Mac Lane knew about complement-toposes via Mortensen. His stance was that they were just the old toposes, but he gave no argument. I have been trying to figure out why did Mac Lane thought that and this is the best I can imagine. I think this would be part of what a mainstream topos- or category-theorist would say at first glance on complement-toposes (and in my experience, this is what they invariably say).
- 35.
Although de Queiroz does not make explicit his use of the last of them. As a side remark, the converse of that theorem does not hold.
- 36.
There is another problem in de Queiroz’s analysis of Brouwerian connectives in a topos. For example, the following diagram
given as definition of Brouwerian subtraction in [11, p. 131] makes no sense. de Queiroz’s definition implies that the equalizer of \(p\vee q\) and the first projection p is \(e_{1} \! : \! \le \! \rightarrowtail \! \Omega \times \Omega \), but it is not correct for when one dualizes
$$(p\supset q) = true \, \text {if and only if} \, (p\wedge q) = p$$one obtains
$$(q-p) = false \, \text {if and only if}\, (p\curlyvee q) = p,$$and not
$$(p-q) = true \, \text {if and only if} \, (p\vee q) = p.$$Briefly, de Queiroz’s definition gives us that Brouwerian subtraction is true in crucial cases where it should be false. Note that one cannot save de Queiroz’s proposal by reverting \(p-q\) as \(q-p\) in his suggested truth condition: In that case the truth conditions of \(\supset \) and \(-\) would coincide and no connective different from \(\supset \), let alone its dual, would have been defined.
- 37.
- 38.
Or bi-Heyting algebras, as has been widely called recently mainly by influence of Reyes and Zolfaghari. See for example [45].
- 39.
A proof that every pair of subobjects in a topos has a union can be found in [8, Proposition 5.10.1]. Here is a sketch. Consider any two subobjects \(i \! : \! : S \! \longrightarrow \! O\) and \(j \! : \! : T \! \longrightarrow \! O\) of O in a topos \(\mathcal {E}\). Take their coproduct \(S+T\) and the morphism \(f \! : \! S+T \! \longrightarrow \! O\). As every morphism in a topos, f factors uniquely into an epimorphism followed by a monomorphism (see [8, Corollary 5.9.4]) \(f = k\circ p \! : \! S+T \! \twoheadrightarrow \! I \! \rightarrowtail \! O\). Morphisms from S and T to I, composed with i and j, imply that these are included in k (I, k is thus the smallest subobject of O which contains both S, i and T, j.) A similar reasoning can be applied to show that every pair of subobjects in a topos has an intersection using products.
- 40.
The next objection can be seen as trying to defend the original definition of negation against the suggested by complement-toposes, but this time using extra-categorial considerations.
- 41.
Yet in [22] it is argued, with [27], that inconsistency-tolerant constructions in sheaf categories in general are not categorically natural (are not preserved by pullbacks along morphisms). A similar point was behind de Queiroz’s objection and the theorems by Reyes et al. I do not know whether there are still hidden “standard” assumptions behind the results in [22]; I will leave the investigation of that for future work.
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Estrada-González, L. (2015). The Evil Twin: The Basics of Complement-Toposes. In: Beziau, JY., Chakraborty, M., Dutta, S. (eds) New Directions in Paraconsistent Logic. Springer Proceedings in Mathematics & Statistics, vol 152. Springer, New Delhi. https://doi.org/10.1007/978-81-322-2719-9_18
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